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Among [[Markov chain Monte Carlo]] (MCMC) [[algorithms]], '''coupling from the past''' is a method for [[sampling (statistics)|sampling]] from the stationary distribution of a [[Markov chain]]. Contrary to many MCMC algorithms, coupling from the past gives in principle a perfect sample from the stationary distribution. It was invented by [[James Propp]] and [[David Bruce Wilson|David Wilson]] in 1996.

==The basic idea==
Consider a finite state [[Markov chain|irreducible]] [[Markov chain|aperiodic]] Markov chain <math>M</math> with state space <math>S</math> and (unique) stationary distribution <math>\pi</math> (<math>\pi</math> is a probability vector). Suppose that we come up with a probability distribution <math>\mu</math> on the set of maps <math>f:S\to S</math> with the property that for every fixed <math>s\in S</math>, its image <math>f(s)</math> is distributed according to the transition probability of <math>M</math> from state <math>s</math>. An example of such a probability distribution is the one where <math>f(s)</math> is [[independent (probability)|independent]] from <math>f(s')</math> whenever <math>s\ne s'</math>, but it is often worthwhile to consider other distributions. Now let <math>f_j</math> for <math>j\in\mathbb Z</math> be independent samples from <math>\mu</math>.

Suppose that <math>x</math> is chosen randomly according to <math>\pi</math> and is independent from the sequence <math>f_j</math>. (We do not worry for now where this <math>x</math> is coming from.) Then <math>f_{-1}(x)</math> is also distributed according to <math>\pi</math>, because <math>\pi</math> is <math>M</math>-stationary and our assumption on the law of <math>f</math>. Define
:<math>F_j:= f_{-1}\circ f_{-2}\circ\cdots\circ f_{-j}.</math>
Then it follows by induction that <math>F_j(x)</math> is also distributed according to <math>\pi</math> for every <math>j\in\mathbb N</math>. Now here is the main point. It may happen that for some <math>n\in\mathbb N</math> the image of the map <math>F_n</math> is a single element of <math>S</math>.
In other words, <math>F_n(x)=F_n(y)</math> for each <math>y\in S</math>. Therefore, we do not need to have access to <math>x</math> in order to compute <math>F_n(x)</math>. The algorithm then involves finding some <math>n\in \mathbb N</math> such that <math>F_n(S)</math> is a [[singleton (mathematics)|singleton]], and outputing the element of that singleton. The design of a good distribution <math>\mu</math> for which the task of finding such an <math>n</math> and computing <math>F_n</math> is not too costly is not always obvious, but has been accomplished successfully in several important instances{{Citation needed|date=May 2008}}.

==The monotone case==
There is a special class of Markov chains in which there are particularly good choices
for <math>\mu</math> and a tool for determining if <math>|F_n(S)|=1</math>. (Here <math>|\cdot|</math> denotes [[cardinal number|cardinality]].) Suppose that <math>S</math> is a [[partially ordered set]] with order <math>\le</math>, which has a unique minimal element <math>s_0</math> and a unique maximal element <math>s_1</math>; that is, every <math>s\in S</math> satisfies <math>s_0\le s\le s_1</math>. Also, suppose that <math>\mu</math> may be chosen to be supported on the set of [[monotone function|monotone]] maps <math>f:S\to S</math>. Then it is easy to see that <math>|F_n(S)|=1</math> if and only if <math>F_n(s_0)=F_n(s_1)</math>, since <math>F_n</math> is monotone. Thus, checking this becomes rather easy. The algorithm can proceed by choosing <math>n:=n_0</math> for some constant <math>n_0</math>, sampling the maps <math>f_{-1},\dots,f_{-n}</math>, and outputing <math>F_n(s_0)</math> if <math>F_n(s_0)=F_n(s_1)</math>. If <math>F_n(s_0)\ne F_n(s_1)</math> the algorithm proceeds by doubling <math>n</math> and repeating as necessary until an output is obtained. (But the algorithm does not resample the maps <math>f_{-j}</math> which were already sampled; it uses the previously sampled maps when needed.)

== References ==
*{{Citation | last1=Propp | first1=James Gary | last2=Wilson | first2=David Bruce | title=Proceedings of the Seventh International Conference on Random Structures and Algorithms (Atlanta, GA, 1995) | mr=1611693 | year=1996 | journal=Random Structures & Algorithms | issn=1042-9832 | volume=9 | issue=1 | chapter=Exact sampling with coupled Markov chains and applications to statistical mechanics | pages=223–252}}
*{{Citation | last1=Propp | first1=James | last2=Wilson | first2=David | title=Microsurveys in discrete probability (Princeton, NJ, 1997) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=DIMACS Ser. Discrete Math. Theoret. Comput. Sci. | mr=1630414 | year=1998 | volume=41 | chapter=Coupling from the past: a user's guide | pages=181–192}}

[[Category:Monte Carlo methods]]
[[Category:Markov chain Monte Carlo]]

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164.67.235.138: use math syntax instead of image

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